Dynamic Repairing A
∗
: A Plan-Repairing Algorithm for Dynamic
Domains
Filippos Gouidis
1,2
, Theodore Patkos
1
, Giorgos Flouris
1
and Dimitris Plexousakis
1,2
1
Institute of Computer Science, FO.R.T.H, Heraklion, Crete, Greece
2
Department of Computer Science, University of Crete, Heraklion, Crete, Greece
Keywords:
Planning, Plan Repairing, Multi-agent Systems, Graph Search.
Abstract:
Re-planning is a special case of planning which arises when already produced plans become invalidated before
their completion. In this work we investigate the conditions under which plan repairing is more efficient than
re-planning from scratch. We present a new plan-repairing algorithm, Dynamic Repairing A
∗
(DRA
∗
), and we
compare its performance against A
∗
in a number of different re-planning scenarios. The experimental results
indicate that if the percentage of the plan that has been already executed is less than 40% to 50% and the
changes in the environment are small or moderate, DRA
∗
outperforms A
∗
in terms of speed by a factor of 10%
to 80% in the majority of the cases.
1 INTRODUCTION
Re-planning is a special case of planning which arises
during the deployment of a plan when either a plan be-
ing deployed no longer satisfies certain criteria (usu-
ally of time or actions’ costs optimality) or some of
its pending actions cannot be executed. In this case,
a new plan has to be produced, and depending on the
way in which this procedure is carried out, the next
two categories can be distinguished: re-planning from
scratch and plan repairing. In the former case, all the
processed information that was used for the produc-
tion of the original plan is discarded, whereas, in the
latter, a part of the previous computational effort is
utilized.
This line of work investigates the conditions un-
der which plan repairing is more efficient than re-
planning from scratch. To this end, we focus our
attention on A
∗
algorithm, which is one of the most
popular and studied algorithms in the field of Artifi-
cial Intelligence. Specifically, our contribution lies in
the development of a novel algorithm, Dynamic Re-
pairing A
∗
(henceforth DRA
∗
), which extends A
∗
in
such a way that it can be used for plan repairing.
Namely, DRA
∗
is suited for the repairing of plans
in dynamic environments and can address modifica-
tions in goal-sets and actions’ costs during the exe-
cution of a plan, which are two of the most common
causes of plan invalidation. Since many state-of-the-
art planning algorithms in a variety of domains are
based on A
∗
the study can provide valuable hints and
insights towards the improvement of the existing re-
planning methods as well as towards the development
of more efficient ones.
Moreover, although it has been demonstrated in
the classic and highly influential work of (Nebel and
Koehler, 1995) that in the worst case modifying an
existing plan is not guaranteed to be more efficient
than re-planning from scratch, the goal of a thorough
understanding regarding the trade-offs between these
two approaches is far from achieved. The current
study wishes to explore in more depth this interac-
tion, revealing practically important instances, where
repairing is guaranteed to be the optimal choice.
The rest of the paper is organized as follows. In
the second section, we describe briefly the A
∗
algo-
rithm. Next, we discuss related work regarding the re-
planning problem. We then present DRA
∗
. In the fifth
section, we continue by presenting our experimental
evaluation comparing DRA
∗
and A
∗
in standard plan-
ning benchmarks. In section 6 we conclude.
2 BACKGROUND
A
∗
(Hart et al., 1968) is one of the most popular
algorithms of Artificial Intelligence, with some of
its most common uses including graph traversal and
path-finding. Its key idea is the utilization of a heuris-
Gouidis, F., Patkos, T., Flouris, G. and Plexousakis, D.
Dynamic Repairing A
∗
: A Plan-Repairing Algorithm for Dynamic Domains.
DOI: 10.5220/0006560803630370
In Proceedings of the 10th International Conference on Agents and Artificial Intelligence (ICAART 2018) - Volume 2, pages 363-370
ISBN: 978-989-758-275-2
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
363

tic value, that “guides” the search. As a result, its per-
formance depends on the quality of the function that
generates the heuristic values.
A
∗
can be implemented in two different ways: a)
using a tree search or b) using a graph search. Typi-
cally, in both cases, two auxiliary collections are uti-
lized during the execution of the algorithm: a priority
queue, called open list, containing the states candidate
for expansion, and a set, referred to as closed list, con-
taining the already expanded states. Moreover, three
special values for each state are used: g-value, h-value
and f-value. The g-value of a state is equal to the cost
from the initial state to it; the h-value is an estimation
of the minimum cost from it to the goal-state and the
f-value is equal to the sum of the g-value and h-value.
At each step of the tree search variation, the state
of the open list having the lowest f-value is removed
from it. The state is examined for satisfying the goal-
set in which case the search stops and the correspond-
ing plan is extracted. Otherwise, the state is expanded
by generating all its successor states which are added
in the open list, while the expanded state is added in
the closed list. If the h-values that are used are con-
sistent
1
, then this variation of the algorithm is guar-
anteed to find an optimal solution, if one exists, and,
moreover, not to generate more states than any other
algorithm that uses the same h-values.
Graph search differs from tree search in two
points. First, each time a state is generated it is ex-
amined for being contained in the closed and open
list respectively. If it is not contained in neither of the
lists, the same steps as in the case of tree search are
followed. If it is already in the closed list, then its
current f-value is compared to its old f-value, e.g. the
one with which it was inserted in the closed list. If
the new f-value is smaller, the state is removed from
the closed list and re-inserted in the open list with the
new f-value.
Moreover, the algorithm in this variation does not
stop when a plan has been found, but it continues until
there is no state in the open list with an f-value that is
smaller than the cost of the plan. In this case, it is not
required that the h-values are consistent, but it suffices
to be admissible, i.e. not to be greater than the cost
of the optimal path from the corresponding state to a
goal-state.
1
The h-value of a state s
N
is consistent, if for every state
s
M
that can be generated from s
N
, the estimated cost of
reaching a goal-state from s
N
is not greater than the cost
of getting to s
M
from s
N
plus the estimated cost of reaching
a goal-state from s
M
.
3 RELATED WORK
Over the last years, a significant number of A
∗
-
inspired plan repairingalgorithms has been devel-
oped, with the majority of them tailored to single-
agent robotics problems. These algorithms fall, typ-
ically, into two main categories w.r.t. their capaci-
ties for plan-repairing: a) algorithms that are special-
ized in addressing modifications of the original goal-
set (Stentz et al., 1995; Koenig and Likhachev, 2002;
Likhachev et al., 2003; Hansen and Zhou, 2007)
and b) algorithms that are specialized in addressing
changes of the actions costs (Koenig et al., 2004; Van
Den Berg et al., 2006; Koenig and Likhachev, 2006).
Finally, there are few other algorithms that can cope
with both changes (Sun et al., 2008; Sun et al., 2010a;
Sun et al., 2010b).
In general, the efficiency of these algorithms de-
rives from the exploitation of the geometrical prop-
erties of the terrain where the agent is situated, since
in some single-agent settings, such as navigation or
moving-target search, the search tree can be mapped
to the problem terrain. However, this mapping cannot
be realized in many single-agent settings or in a multi-
agent environment and, as a consequence, these algo-
rithms are not applicable for problems of this type.
Two of the most influential algorithms of the first
category are, Focused D
∗
(Stentz et al., 1995), and
D
∗
-Lite (Koenig and Likhachev, 2002). Both of them
utilized a backwards-directed search from the goal
state to the current state, saving, this way, informa-
tion, which allows fast plan production when changes
in the environment occur.
The Generalized Adaptive A
∗
(GAA
∗
) is pre-
sented in (Sun et al., 2008). GAA
∗
learns h-values
in order to make them more informed and can be uti-
lized for moving target search in terrains where the ac-
tion costs of the agent can change between searches.
An extension of GAA
∗
that is close to our work, is
MP-GAA
∗
(Hern
´
andez et al., 2015), where some of
the best paths for some nodes of the search graph are
stored. More recently, there have been implemented
Generalized Fringe-Retrieving A
∗
(Sun et al., 2010a)
and Moving Target D
∗
-Lite (Sun et al., 2010b) which,
in the same way as GAA
∗
can address both goal-set
modifications and actions costs changes.
Finally, we mention briefly some other plan re-
pairing approaches that are not based on A
∗
and where
ad hoc techniques such as plan refinement and adap-
tation are utilized. For example, in (Gerevini and
Serina, 2010) specialized heuristic search techniques
are used in order to solve the plan adaptation tasks
through the repairing of certain portions of the origi-
nal plan. Similarly, in (Au et al., 2002) a special algo-
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
364

rithm which uses analogy by derivation for plan adap-
tation is presented. However, in contrast with DRA
∗
,
in these cases plan optimality is not a central issue.
4 DYNAMIC REPAIRING A
∗
∗
∗
DRA
∗
is an extension of A
∗
that is suited for the re-
pairing of sequential plans and can address two types
of changes in the environment: a) goal-set modifica-
tions and b) actions’ costs alterations. DRA
∗
is based
on the graph search variation of A
∗
, utilizing the same
search strategy: the selection, testing and expansion
of a state at each step and the utilization of a heuristic
value to guide the whole procedure.
Its novelty is that a new search graph is not created
from scratch as in A
∗
. Instead, the initial search graph
is retrieved at the start of the algorithm, and then used
for the subsequent search. As with the case of graph
search A
∗
, the utilization of admissible h-values is re-
quired in order for the solutions returned to be opti-
mal. The corresponding pseudo-code is presented in
pages 3 and 4.
4.1 Comparing DRA
∗
∗
∗
with A
∗
∗
∗
DRA
∗
differs from A
∗
in a number of ways. First,
while in the case of A
∗
only the parent state, i.e. the
state that results in the lowest g-value, is kept, in the
case of DRA
∗
every predecessor state of a given state
is stored. Moreover, a special procedure, the inform-
ing procedure, takes place, in order to be determined
if a state that is derived from a previous planning pro-
cedure and is encountered for the first time, is reach-
able from the new initial state, and its g-value and h-
value to be updated if necessary. As a consequence,
the novel concepts of an informed, uninformed, valid
and invalid state are introduced, which serve for the
description of the corresponding states. By default,
when the algorithm begins, all the states of the search
tree are marked as uninformed except of the new ini-
tial state which is set as informed and valid.
The informing procedure can be achieved in two
different ways: fully or lazily. The former is applied
in cases when there exist actions with decreased costs,
whereas the latter is applied in the other cases.
The procedure for the full informing is the fol-
lowing. First, the state being examined is marked as
pending and, consequently, all its predecessor states
are examined for being informed. For any predeces-
sor state found not to be informed and not to be pend-
ing, the procedure of full informing is followed. If
Algorithm 1: Dynamic Repairing A
∗
.
input : New Initial State, Previous Closed List, Previous Open
List, Original Goal set, New Goal set
output: The optimal plan for the new goal set
1 plan ←− NU LL
2 mark newInitialState as valid and informed
3 CLOSED ←− previousCLOSED
4 if originalGoalSet = newGoalSet then
5 plan ←− searchCloseList(CLOSED,newGoalSet)
6 if plan 6= NULL & @ action with decreased costs then
7 return plan
8 OPEN ←− previousOPEN
9 if newGoalSet is not superset o f originalGoalSet then
10 validateOpenList(OPEN,originalGoalSet,newGoalSet)
11 while OPEN is not empty do
12 currentState ←− OPEN.poll()
13 if currentState satisfies newGoalSet then
14 plan ←− ExtractPlan(currentState)
15 break
16 foreach applicable action ac of currentState do
17 succState ←− currentState.apply(ac)
18 pVal ←− currentState.gValue +ac.cost
19 if succState /∈ OPEN and /∈ CLOSED then
20 OPEN.add(succState)
21 else
22 if succState is not informed then
23 lazy inform(succState)
24 if succState is not valid then
25 OPEN.add(succState)
26 else
27 if pVal < succState.gValue then
28 if OPEN 3 succState then
29 OPEN.remove(succState)
30 if CLOSED 3 succStatet then
31 CLOSED.remove(succState)
32 OPEN.add(succState)
33 else
34 succState.predQueue.add(currentState
)
35 CLOSED.add(currentState )
36 end
37 end
38 OPEN.add(plan.currentState )
39 previousOPEN ←− OPEN
40 previousCLOSED ←− CLOSED
41 return plan
Algorithm 2: Validation of the Open List.
input : Open List, Original Goal Set and New Goal Set
1 newOpenList ←− new Priority Queue()
2 foreach state in OPEN do
3 if state is not In f ormed then
4 if @ action with decreased costs then
5 lazy inform(state)
6 else
7 full infrom(state)
8 if state is Valid then
9 state.update fValue()
10 newOpenList.add(state)
11 end
12 OPEN ←− newOpenList
Dynamic Repairing A
∗
: A Plan-Repairing Algorithm for Dynamic Domains
365

Algorithm 3: Traversal of the Closed List.
input : Closed List and New Goal Set
output: A plan
1 plan ←− null
2 cost = ∞
3 foreach State state in CLOSED do
4 if state satis f ies goalSet then
5 if state is not In f ormed then
6 lazy inform(state);
7 if state is Valid then
8 if state.gValue < cost then
9 plan ←− ExtractPlan(state)
10 cost =state.gValue
11 end
12 return plan
Algorithm 4: Successor States Update.
input : A state stateUpd
1 if stateUpd is informed and valid then
2 foreach successor state in stateU pd.StatesList do
3 ac ←− successor state.generatingAction
4 pVal ←− stateU pd.gValue + ac.cost
5 successor state.nonInformedPredecessors--
6 if pVal < successor state.gValue then
7 successor state.gValue = pVal
8 genState.parent ←− stateU pd
9 if successor state.nonInformedPredecessors=0
then
10 marked successor state as informed and
valid
11 successor state.updatesuccStates()
12 end
13 else
14 foreach successor state in stateU pd.StatesList do
15 successor state.nonInformedPredecessors--
16 if successor state.nonInformedPredecessors=0 then
17 marked successor state as informed and invalid
18 successor state.updatesuccStates()
19 end
a valid predecessor states is found,its p-value
2
is re-
calculated and compared with the state’s g-value and
if found smaller, then this predecessor state is set as
the state’s parent. After the examination of the pre-
decessor states finishes, if the state has any valid pre-
decessor, it is marked as informed and valid. Finally,
the state is reset from pending.
The lazy informing (the corresponding pseu-
docode is omitted due to lack of space) is carried out
in the same way as the full, with the exception that
the procedure stops if a valid predecessor state has
been found and the next predecessor state that is to be
examined does not have a smaller p-value. The exam-
ination of the predecessor states follows their sorting
order. That is, it begins with predecessor state hav-
ing the lowest p-value, i.e. the parent-state, and con-
tinues with the one having the second lowest and so
forth. Note that in this case, some of the p-values of a
2
With the term p-value
s
a
→ s
d
we denote the resulting g-
value of s
d
after its generation of another state s
a
.
Algorithm 5: Full informing.
input : A state stateInf
1 set stateIn f as pending
2 parentState ←− stateIn f .getParent()
3 stateIn f .gValue =∝
4 nonIn f ormedPredecessors = 0
5 if parentState not in f ormed & not pending then
6 full inform(parentState)
7 if parentState is Valid then
8 stateIn f .gValue = parentState.gValue + action.cost
9 if parentState is pending then
10 nonIn f ormedPredecessors + +
11 parentState.StatesList.add(stateInf )
12 if parentState.nonIn f ormedPredecessors > 0 then
13 nonIn f ormedPredecessors + +
14 parentState.StatesList.add(stateInf )
15 foreach predState in stateInf .predQueue do
16 if predState not informed & not pending then
17 full inform(predState )
18 if predState is pending then
19 nonIn f ormedPredecessors + +
20 predState .StatesList.add(stateInf )
21 if predState.nonIn f ormedPredecessors > 0 then
22 nonIn f ormedPredecessors + +
23 predState .StatesList.add(stateInf )
24 if predState is Valid then
25 if predState.pValue < stateIn f .gValue then
26 stateIn f .lgvParent ←− predState
27 stateIn f .gValue ←− predState.pValue
28 end
29 if stateIn f .gValue 6=∝ then
30 mark stateInf as valid and informed
31 else
32 mark stateInf as invalid and informed
33 updatesuccStates(stateInf )
34 reset stateInf from pending
state might not be correct and some of its parent states
might not have been informed, without this affecting
the correctness of the algorithm.
Another difference between the two algorithms is
that in the case of DRA
∗
, when the search for the plan
finishes, the closed and open lists are stored, so that
they can be used in case of re-planning. Before the
open list is saved, the last removed state is re-inserted
in it. Subsequently, when the algorithm is executed,
the previously save lists are retrieved and used.
In addition, if the new goal set is the same as
the original goal set, the initial closed list is searched
for containing solutions before the main part of the
algorithm begins. During this traversal, each state
is examined. The ones satisfying the new goal-set
are lazily informed, when uninformed. In case one
or more valid states satisfying the new goal-set have
been found, the one having the lowest g-value is re-
turned as solution and the algorithm terminates.
Finally, in cases where the new goal set is not a
superset of the original goal set the open list is val-
idated before the main search starts. Namely, every
state is informed, fully if there exists actions with de-
creased costs and lazily otherwise, and, if it is valid,
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
366

its h-value is re-calculated and it is re-inserted in the
open list with its newly updated f-value.
Theorem 1. DRA
∗
is sound and complete for re-
pairing scenarios of goal-set modifications or actions
costs changes if the h-values that are used are admis-
sible
3
.
5 EXPERIMENTAL EVALUATION
For the assessment of the capacity of DRA
∗
in ad-
dressing re-planning problems, we compared its per-
formance in terms of speed against A
∗
. To this end,
we devised four different re-planning scenarios sim-
ulating real-world situations, where we compared the
ratio of the runtime of the two algorithms by varying
the following characteristics:
• The percentage of the original plan that was al-
ready executed at the time when the need for re-
planning occurred (Scenarios 1, 2, 3 and 4);
• The percentage of the modification of the original
goal-set (Scenarios 1 and 2);
• The percentage of the actions whose costs de-
creased (Scenario 3);
• The percentage of the actions whose costs in-
creased (Scenario 4).
We opted for comparing DRA
∗
against A
∗
instead
of other replanning algorithms for two reasons. First,
A
∗
is the most typical planning algorithm, the prop-
erties and behavior of which have been thoroughly
studied. Therefore, the experimental results concern-
ing the relative performance of the two algorithms,
could provide us with valuable hints and insights for
a better understanding of DRA
∗
. Second, most of
the re-planning algorithms, as we already mentioned
in section 3, are utilizable only in specific settings,
which usually concern single-agent problems, and,
therefore, are not applicable in the scenarios we are
examining.
The experiments focus on time performance;
memory requirements are not measured, because both
A
∗
and DRA
∗
exhibit a linear complexity in the num-
ber of states in the state space. We should note, also,
that we did not include in the experimental results
the generated and expanded states of each algorithm
since DRA
∗
always expands and generates less states
than A
∗
when a part of the search tree is already con-
structed which is the case in all the conducted experi-
ments.
3
The proof is presented in (Gouidis et al., 2017).
5.1 Experimental Setup
The structure of the experiments is the same in every
case. First, a plan is produced for the initial condi-
tions of the problem, i.e. initial state, goal-set and
actions’ costs. Next, a parameter of the environment,
according to the type of the experiment, is modified:
in scenarios 1 and 2 the goal-set, and in scenarios
3 and 4 the costs of some actions. Finally, a new
plan is produced for the modified conditions using
both A
∗
(replanning from scratch) and DRA
∗
(repair-
ing). The new initial state of the re-planning problems
is a randomly-selected state of the initial plan. The
changes for each scenario are the following:
• Scenario 1. The new goal set is produced by the
removal of k goals from the initial goal set con-
sisted of n goals, and the insertion of m goals in it
respectively, where k ≤ n.
• Scenario 2. The new goal set is produced by the
addition of k goals in the original goal-set.
• Scenario 3. A p% percentage of the actions costs
are decreased, none of which belongs to the initial
plan. The maximum decrease for an action cost is
a 90% of its initial cost.
• Scenario 4. A p% percentage of the actions costs
are increased. q% of the actions with increased
costs belongs to the initial plan. The maximum
increase for an action cost is a 200% of its initial
cost.
The benchmarks that were used for the evalua-
tion are: Blocks, Depots, Gripper, Logistics, Miconic
and Transports which derive from the 3
rd
, 4
th
and 8
th
International Planning Competitions (Bacchus, 2001;
Long and Fox, 2003; Gerevini et al., 2009). In addi-
tion, since in the majority of the planning domains the
actions costs are uniform, we created two variations
of the domains Logistics and Depots, Logistics-cost
and Depots-cost respectively, with actions of varied
costs. The specifications of the scenarios are shown
in Table 1.
Both algorithms were implemented in Java, us-
ing the same data structures, functions and routines
for all the shared procedures, in order to ensure that
the disparities in the runtimes reflect performative
differences between the algorithms and are not due
to their different implementations. The experiments
were conducted on a 64-bit Ubuntu Workstation with
two 8-core
R

Xeon
R

CPU E5-2630 processors run-
ning at a 2.30GHz server with 384 GB RAM, from
which 10 GB were allocated for each experiment.
5.2 Experimental Results
The experimental results, presented in Table 2, indi-
Dynamic Repairing A
∗
: A Plan-Repairing Algorithm for Dynamic Domains
367

Table 1: Specifications of the scenarios’ experiments.
Experiment Problem
Number of
Initial Goals
Number of
Removed Goals
Number of
Added Goals
Average Branching
Factor
Number of
Conducted
Experiments
1.1 Blocks 6 1 1 4.61 5
1.2 Depots 5 1 1 8.77 5
1.3 Gripper 12 2 3 4.62 5
1.4 Logistics 5 1 1 8.33 5
1.5 Logistics 5 1 1 8.44 5
2.1-2.4 Blocks 6/7/6/7 2/1/2/1 - 4.92/4.95/4.62/4.69 28/28/28/28
2.5-2.8 Logistics 4/5/5/4 2/1/2/1 - 8.38/8.46/8.51/8.49 15/6/15/6
2.9-2.12 Depot 3/4/3/4 2/1/3/4 - 10.88/10.84/4.62/4.65 10/5/20/15
2.13-2.15 Gripper 8/9/11 4/5/3 - 4.38/4.35/4.66 40/40/40
2.16-2.19 Miconic 7/10/11/12 3/1/2/4 - 19.83/19.97/21.71/23.67 10/10/10/10
Experiment Problem
Percentage of
Decreased (Increased)
Actions Costs
Max Percentage of Plan’s
Decreased (Increased)
Actions Costs
Average
Branching Factor
Number of
Conducted
Experiments
3.1a/3.1b/3.1c Transport 5/25/50 90 8.93 10
3.2a/3.2b/3.2c Depots-cost 5/25/50 90 4.62 10
3.3a/3.3b/3.3c Logistic-cost 5/25/50 90 8.23 10
4.1a/4.1b/4.1c Transport 5/25/50 200 8.94 10
4.2a/4.2b/4.2c Depots-cost 5/25/50 200 4.57 10
4.3a/4.3b/4.3c Logistic-cost 5/25/50 200 8.16 10
cate that DRA
∗
outperforms A
∗
in most of the goal
set modification cases, provided that the next condi-
tions are met. First, the percentage of the original
plan that has been already executed, should not be
greater than 50%. Moreover, the change in the goal-
set should not be greater than 20% to 50%. The corre-
sponding thresholds, for the previous two parameters,
below which DRA
∗
performs better, depend on the
average branching factor of the re-planning problem,
with average higher branching factors corresponding
to thresholds of lower values. Moreover, according
to the results, in the cases of modified actions costs,
DRA
∗
outperforms A
∗
always.
Furthermore, we can make the following observa-
tions regarding the performance of DRA
∗
compared
to A
∗
:
1. As the percentage of the executed plan decreases,
the relative performance is improved.
2. As the percentage of the modified goal-set de-
creases, the relative performance is improved.
3. As the average branching factor, i.e. the average
number of predecessor states that a state has, de-
creases, the relative performance is improved.
4. The relative performance does not vary signifi-
cantly as the percentage of actions with decreased
costs increases.
5. The relative performance does not vary signifi-
cantly as the percentage of actions with increased
costs increases.
6. For a given problem instance, DRA
∗
performs bet-
ter in increases of the goal-set than in general
modifications of the goal-set.
7. For a given problem instance, DRA
∗
performs bet-
ter in cases of increased actions costs than
of decreased actions costs.
We consider that the previous findings can be ex-
plained by the following reasons. First, DRA
∗
ex-
pands at most the same number of states as A
∗
, since a
part of the search graph with which the search begins,
is already constructed. Moreover, during DRA
∗
exe-
cution, the procedures of states informing, open list
validation and closed list traversal, which are absent
from A
∗
, might take place. Therefore, it can be con-
cluded, that the trade-off between the previous two
factors determines DRA
∗
performance against A
∗
.
Regarding the first finding, it can be due to the fact
that as the percentage of the executed plan increases,
the new root of the search graph recedes further from
the root of the original search graph, which, as a re-
sult, has one of the following two outcomes: a larger
part of the search graph leaves would either become
invalid or would have its f-values increased. In either
case, time is consumed for the informing of states that
do not affect the search.
Likewise, the fact that the traversal of the closed
list and the validation of the open list, is not carried
out in the case of an increased goal-set seems to ex-
plain the better performance of DRA
∗
in such cases in
comparison to the general case of handling modified
goal-sets (observation 6). A similar line of reasoning
can be applied in the case of modified actions costs
(observation 7). Namely, in the case of decreased
costs, the open list is validated. Furthermore, the in-
forming of the states is full, whereas, in the case of
increased costs, the lazy informing is utilized, which,
at worst case, requires the same time. Findings 4 and
5 can be ascribed to the fact that greater percentages
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
368

Table 2: Mean value of the ratio of the runtime of DRA
∗
to A
∗
for scenarios 1-4.
Percentage of Executed Plan
Experiment 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Scenario 1
1.1 0.85±0.10 0.83±0.21 0.78±0.14 0.76±0.19 0.97±0.21 0.94±0.23 1.23±0.22 1.12±0.31 0.63±0.28 1.2 0.44±0.10 0.44±0.05 0.61±0.05 1.33±0.05 1.78±0.35 2.58±0.31 2.95±0.32 3.21±0.41 4.04±0.54 1.3 0.22±0.05 0.23±0.03 0.21±0.06 0.24±0.03 1.21±0.53 1.86±1.01 2.30±1.62 2.92±1.57 3.36±2.01 1.4 0.41±0.07 0.29±0.06 0.58±0.09 1.36±0.16 1.28±0.14 2.70±0.37 4.82±0.45 5.67±0.64 7.03±0.82 1.5 0.23±0.03 0.32±0.13 0.53±0.14 1.17±0.21 1.52±0.94 1.68±0.00 2.92±1.34 4.20±1.76 6.04±2.54
Scenario 2
2.1 0.85±0.09 0.84±0.11 0.81±0.12 0.80±0.13 0.88±0.13 0.94±0.12 0.99±0.06 0.98±0.08 0.92±0.30 2.2 0.65±0.06 0.62±0.06 0.59±0.06 0.48±0.13 0.77±0.15 1.07±0.17 0.93±0.08 1.08±0.15 0.85±0.08 2.3 0.85±0.14 0.84±0.15 0.78±0.13 0.78±0.19 0.77±0.14 1.47±1.23 1.01±0.33 2.85±2.58 1.72±1.53 2.4 0.67±0.13 0.60±0.17 0.59±0.17 0.60±0.27 0.69±0.18 2.53±1.89 1.29±0.71 1.66±1.42 1.65±1.15 2.5 0.96±0.16 0.96±0.19 0.90±0.18 0.86±0.13 0.99±0.12 1.10±0.25 1.89±1.62 2.47±1.32 3.19±1.66 2.6 0.60±0.26 0.59±0.24 0.70±0.13 0.75±0.10 2.06±2.14 2.99±2.36 3.46±1.76 4.50±1.93 6.10±2.44 2.7 0.99±0.12 1.00±0.10 0.96±0.13 0.93±0.15 0.96±0.10 1.03±0.09 1.39±0.46 2.59±1.53 3.33±2.23 2.8 0.58±0.22 0.55±0.20 0.57±0.15 0.65±0.21 0.85±0.14 1.64±0.82 3.54±0.88 4.73±1.35n 5.74±1.66 2.9 0.87±0.15 0.85±0.17 0.82±0.14 0.86±0.16 0.82±0.13 1.01±0.07 2.16±1.22 3.63±2.27 4.27±2.06 2.1 0.57±0.02 0.51±0.09 0.49±0.06 0.45±0.17 1.09±0.81 1.09±0.89 1.50±1.23 2.42±1.76 3.45±2.03
2.11 0.83±0.04 0.85±0.06 0.79±0.08 0.83±0.07 0.84±0.08 0.82±0.04 1.01±0.17 2.64±1.18 3.41±0.34 2.12 0.65±0.19 0.64±0.21 0.60±0.20 0.59±0.18 0.71±0.16 1.40±1.49 2.78±1.64 3.65±2.63 3.87±1.96 2.13 0.48±0.04 0.46±0.04 0.46±0.03 0.45±0.04 0.43±0.04 0.44±0.05 0.49±0.06 0.74±0.15 1.26±0.29 2.14 0.80±0.14 0.78±0.09 0.73±0.11 0.69±0.08 0.71±0.09 0.77±0.07 0.87±0.08 1.37±0.38 2.75±1.83 2.15 0.32±0.02 0.30±0.01 0.28±0.01 0.29±0.02 0.26±0.05 0.56±0.11 1.10±0.17 3.60±0.19 4.56±0.25 2.16 0.53±0.15 0.72±0.10 1.44±0.35 2.11±0.86 4.01±0.70 5.10±1.46 6.75±0.00 8.43±3.21 10.03±3.42 2.17 0.21±0.05 0.54±0.15 0.95±0.05 2.08±0.57 4.85±1.76 5.72±1.87 6.44±2.22 8.32±2.42 9.84±3.27 2.18 0.17±0.03 0.63±0.07 1.03±0.34 3.70±0.60 5.30±1.71 6.90±2.41 7.72±2.74 9.24±3.03 10.11±2.87 2.19 0.62±0.13 1.13±0.60 1.56±0.99 2.68±1.04 4.43±1.44 6.10±2.54 8.68±3.04 9.52±3.04 11.30±3.32
Scenario 3
3.1a 0.43±0.13 0.68±0.17 0.73±0.14 0.59±0.16 0.64±0.17 0.83±0.21 0.72±0.21 0.85±0.23 0.81±0.27
3.1b 0.50±0.12 0.54±0.14 0.59±0.14 0.62±0.21 0.54±0.22 0.69±0.25 0.69±0.22 0.92±0.26 0.78±0.25
3.1c 0.49±0.15 0.55±0.16 0.53±0.19 0.70±0.22 0.64±0.24 0.62±0.22 0.71±0.25 0.90±0.25 0.84±0.26 3.2a 0.17±0.03 0.18±0.04 0.24±0.04 0.22±0.05 0.24±0.06 0.21±0.07 0.20±0.06 0.24±0.08 0.30±0.07
3.2b 0.25±0.05 0.29±0.05 0.23±0.04 0.29±0.06 0.34±0.07 0.31±0.09 0.30±0.09 0.35±0.10 0.32±0.11
3.2c 0.26±0.07 0.32±0.06 0.28±0.06 0.29±0.07 0.37±0.04 0.39±0.11 0.35±0.12 0.36±0.11 0.39±0.12 3.3a 0.41±0.08 0.44±0.07 0.56±0.11 0.54±0.13 0.46±0.14 0.48±0.18 0.52±0.19 0.61±0.21 0.59±0.20
3.3b 0.45±0.08 0.46±0.08 0.48±0.12 0.50±0.15 0.54±0.18 0.52±0.17 0.61±0.22 0.60±0.24 0.59±0.22
3.3c 0.49±0.10 0.56±0.12 0.52±0.14 0.62±0.16 0.55±0.17 0.55±0.19 0.69±0.24 0.65±0.23 0.68±0.25
Scenario 4
4.1a 0.21±0.03 0.23±0.07 0.19±0.34 0.25±0.07 0.30±0.10 0.65±0.15 0.77±0.14 0.64±0.13 0.80±0.17
4.1b 0.48±0.13 0.62±0.14 0.75±0.19 0.79±0.18 0.87±0.19 0.79±0.21 0.77±0.20 0.73±0.24 0.85±0.23
4.1c 0.54±0.13 0.50±0.16 0.65±0.19 0.68±0.24 0.73±0.24 0.70±0.25 1.00±0.28 0.83±0.26 0.84±0.22 4.2a 0.10±0.03 0.08±0.04 0.12±0.04 0.10±0.06 0.24±0.11 0.35±0.14 0.27±0.13 0.28±0.13 0.30±0.12
4.2b 0.09±0.04 0.13±0.05 0.13±0.06 0.18±0.07 0.26±0.10 0.25±0.12 0.27±0.11 0.37±0.14 0.33±0.13
4.2c 0.21±0.10 0.27±0.13 0.25±0.11 0.35±0.17 0.30±0.15 0.29±0.16 0.37±0.16 0.35±0.15 0.39±0.13 4.3a 0.14±0.05 0.24±0.09 0.39±0.14 0.45±0.16 0.37±0.17 0.34±0.14 0.28±0.14 0.43±0.13 0.36±0.18
4.3b 0.26±0.05 0.33±0.11 0.38±0.12 0.48±0.13 0.41±0.15 0.44±0.15 0.42±0.16 0.39±0.17 0.45±0.15
4.3c 0.41±0.08 0.54±0.15 0.43±0.16 0.67±0.15 0.54±0.18 0.70±0.19 0.75±0.21 0.80±0.22 0.74±0.21
of modified actions costs do not affect the execution
of DRA
∗
.
Finally, the deterioration of DRA
∗
performance
with higher average branching factors (observation 3)
can be attributed to the greater time that is neces-
sary for the informing procedure, since large average
branching factors correspond to a large average num-
ber of predecessor states, which results in a greater
number of examined ancestor states during the in-
forming procedure.
6 CONCLUSIONS
In this work we presented a novel plan repairing al-
gorithm, DRA
∗
, that extends one of the most popular
and studied planning algorithms, A
∗
, by addressing
modifications in the goal-set and in the actions costs.
Therefore, DRA
∗
, is suitable for plan repairing in dy-
namic environments, where changes of the aforemen-
tioned kinds take place.
The conducted experimental evaluation showed
Dynamic Repairing A
∗
: A Plan-Repairing Algorithm for Dynamic Domains
369

that DRA
∗
outperformed A
∗
in most of the cases with
modified goal-sets, provided that the percentage of
the original plan that has been already executed, is not
greater than 40% to 50%. and the change in the goal-
set is not be greater than 20% to 50%. The overall
performance depends on the average branching factor
of the problem, with average higher branching factors
corresponding to thresholds of lower values. For re-
planning scenarios of modified actions costs, the ex-
perimental outcome was that DRA
∗
outperformed A
∗
in all experiments.
We believe that the experimental results provide
a strong support for the utilization of DRA
∗
in re-
planning scenarios. Nevertheless, a more thorough
experimental analysis could provide more useful hints
and insights and help us to gain a more elaborate
understanding of the underlying mechanisms which
determine the strengths and weaknesses of the algo-
rithm.
In particular, we would like to assess DRA
∗
perfor-
mance in scenarios of repeated repairing and in sce-
narios where both the goal-set and the actions costs
are modified, which seem to represent more faithfully
certain dynamic environments. Another direction that
we wish to investigate is the addressing of other types
of dynamicity that can be observed in real-world do-
mains, such as altered preconditions and effects for
actions, additions and removals of planning agents
and invalidations or insertions of new actions.
Moreover, since the worst performance of DRA
∗
is observed in domains with large branching factors
which are directly related to the number of the agents
activated for the re-planning procedure, we consider
that a distributed implementation of DRA
∗
, where
each agent performs an independent search, could im-
prove substantially the performance of the algorithm.
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